Verfügbarkeit: A concrete approach to classical analysis

A concrete approach to classical analysis:

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Bibliographische Detailangaben
1. Verfasser:	Mureşan, Marian ca. 20./21. Jh (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	New York, NY Springer 2009
Schriftenreihe:	CMS books in mathematics
Schlagworte:	Calculus, Integral Differential calculus Mathematical analysis Differentialrechnung Analysis Integralrechnung
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XVIII, 433 S. graph. Darst. 235 mm x 155 mm
ISBN:	0387789324 9780387789323

Internformat

MARC


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Datensatz im Suchindex

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adam_text	Contents Sets and Numbers ......................................... 1 1.1 Sets ................................................... 1 1.1.1 The concept of a set ............................... 1 1.1.2 Operations on sets ................................. 3 1.1.3 Relations and functions ............................ 6 1.2 Sets of numbers ......................................... 11 1.2.1 Two examples .................................... 11 1.2.2 The real number system ............................ 12 1.2.3 Elements of algebra ................................ 20 1.2.4 Elements of topology on Ж ......................... 25 1.2.5 The extended real number system ................... 30 1.2.6 The complex number system ........................ 31 1.3 Exercises ............................................... 32 1.4 References and comments ................................ 40 Vector Spaces and Metric Spaces .......................... 41 2.1 Vector spaces ........................................... 41 2.1.1 Finite-dimensional vector spaces ..................... 41 2.1.2 Vector spaces ..................................... 44 2.1.3 Normed spaces .................................... 48 2.1.4 Hubert spaces ..................................... 49 2.1.5 Inequalities ....................................... 51 2.2 Metric spaces ........................................... 57 2.3 Compact spaces ......................................... 65 2.4 Exercises ............................................... 70 2.5 References and comments ................................ 70 Sequences and Series ...................................... 73 3.1 Numerical sequences ..................................... 73 3.1.1 Convergent sequences .............................. 73 3.1.2 Subsequences ..................................... 76 Contents 3.1.3 Cauchy sequences ................................. 77 3.1.4 Monotonie sequences ............................... 80 3.1.5 Upper limits and lower limits ....................... 81 3.1.6 The big Oh and small oh notations .................. 82 3.1.7 Stolz-Cesaro theorem and some of its consequences .... 85 3.1.8 Certain combinatorial numbers ...................... 88 3.1.9 Unimodal, log-convex, and Pólya-frequency sequences . . 94 3.1.10 Some special sequences ............................. 98 3.2 Sequences of functions ...................................108 3.3 Numerical series .........................................110 3.3.1 Series of nonnegative terms .........................113 3.3.2 The root and the ratio tests ........................123 3.3.3 Partial summation .................................125 3.3.4 Absolutely and conditionally convergent series ........126 3.3.5 The W-Z method ................................130 3.4 Series of functions .......................................134 3.4.1 Power series ......................................137 3.4.2 Hyp er geometric series ..............................138 3.5 The Riemann Zeta function ζ(ρ) ..........................139 3.6 Exercises ...............................................139 3.7 References and comments ................................144 Limits and Continuity .....................................147 4.1 Limits .................................................147 4.1.1 The limit of a function .............................147 4.1.2 Right-hand side and left-hand side limits .............152 4.2 Continuity ..............................................152 4.2.1 Continuity and compactness ........................155 4.2.2 Uniform continuous mappings .......................156 4.2.3 Continuity and connectedness .......................160 4.2.4 Discontinuities ....................................161 4.2.5 Monotonie functions ...............................161 4.3 Periodic functions .......................................164 4.4 Darboux functions .......................................165 4.5 Lipschitz functions .......................................167 4.6 Convex functions ........................................169 4.6.1 Convex functions ..................................169 4.6.2 Jensen convex functions ............................173 4.7 Functions of bounded variations ...........................177 4.8 Continuity of sequences of functions .......................183 4.9 Continuity of series of functions ...........................184 4.10 Exercises ...............................................186 4.11 References and comments ................................189 Contents xi Differential Calculus on К .................................191 5.1 The derivative of a real function ...........................191 5.2 Mean value theorems ....................................197 5.2.1 Consequences of the mean value theorems ............201 5.3 The continuity and the surjectivity of derivatives ............206 5.4 L Hospital theorem ......................................207 5.5 Higher-order derivatives and the Taylor formula .............208 5.6 Convex functions and differentiability ......................214 5.6.1 Inequalities .......................................216 5.7 Differentiability of sequences and series of functions ..........217 5.8 Power series and Taylor series .............................219 5.8.1 Operations with power series ........................222 5.8.2 The Taylor expansion of some elementary functions .... 225 5.8.3 Bernoulli numbers and polynomials ..................228 5.9 Some elementary functions introduced by recurrences ........230 5.9.1 The square root function ...........................231 5.9.2 The logarithm function .............................231 5.9.3 The exponential function ...........................235 5.9.4 The arctangent function ............................237 5.10 Functions with primitives .................................239 5.10.1 The concept of a primitive function ..................239 5.10.2 The existence of primitives for continuous functions .... 242 5.10.3 Operations with functions with primitives ............244 5.11 Exercises ...............................................247 5.12 References and comments ................................249 Integral Calculus on E ....................................251 6.1 The Darboux-Stieltjes integral ............................251 6.1.1 The Darboux integral ..............................251 6.1.2 The Darboux-Stieltjes integral ......................252 6.2 Integrability of sequences and series of functions .............262 6.3 Improper integrals .......................................263 6.4 Euler integrals ..........................................271 6.4.1 Gamma function ..................................271 6.4.2 Beta function .....................................275 6.5 Polylogarithms ..........................................278 6.6 e and π are transcendental ...............................280 6.7 The Grönwall inequality ..................................283 6.8 Exercises ...............................................284 6.9 References and comments ................................287 Contents Differential Calculus on Ж ................................289 7.1 Linear and bounded mappings ............................289 7.1.1 Multilinear mappings ..............................293 7.1.2 Quadratic mappings ...............................294 7.2 Differentiable functions ...................................296 7.2.1 Variations ........................................296 7.2.2 Gâteaux differential ...............................297 7.2.3 Fréchet differential ................................298 7.2.4 Properties of the Fréchet differentiable functions ......300 7.3 Partial derivatives .......................................304 7.3.1 The inverse function theorem and the implicit function theorem ..........................................307 7.3.2 Directional derivatives and gradients .................312 7.4 Higher-order differentials and partial derivatives .............312 7.4.1 The case X = Жп .................................314 7.5 Taylor formula ..........................................316 7.6 Problems of local extremes ...............................317 7.6.1 First-order conditions ..............................317 7.6.2 Second-order conditions ............................318 7.6.3 Constraint local extremes ..........................319 7.7 Exercises ...............................................322 7.8 References and comments ................................322 Double Integrals, Triple Integrals, and Line Integrals ......325 8.1 Double integrals .........................................325 8.1.1 Double integrals on rectangles .......................325 8.1.2 Double integrals on simple domains ..................331 8.2 Triple integrals ..........................................333 8.2.1 Triple integrals on parallelepipeds ...................333 8.2.2 Triple integrals on simple domains ...................340 8.3 и -fold integrals .........................................341 8.3.1 Ti-fold integrals on hyperreet angles ..................341 8.3.2 n-fold integrals on simple domains ..................345 8.4 Line integrals ...........................................346 8.4.1 Line integrals with respect to arc length ..............346 8.4.2 Line integrals with respect to axis ...................347 8.4.3 Green formula ....................................347 8.5 Integrals depending on parameters .........................349 8.6 Exercises ...............................................353 8.7 References and comments ................................354 Contents xiii 9 Constants .................................................355 9.1 Pythagoras s constant ....................................355 9.1.1 Sequences approaching л/2 ......................... 355 9.2 Archimedes constant ....................................356 9.2.1 Recurrence relation ................................356 9.2.2 Buffon needle problem .............................358 9.3 Arithmetic-geometric mean ...............................358 9.4 BBP formulas ...........................................363 9.4.1 Computing the nth binary or hexadecimal digit of π . . 363 9.4.2 BBP formulas by binomial sums .....................368 9.5 Ramanujan formulas .....................................372 9.6 Several natural ways to introduce number e ................374 9.7 Optimal stopping problem ................................377 9.8 References and comments ................................378 10 Asymptotic and Combinatorial Estimates ..................381 10.1 Asymptotic estimates ....................................381 10.2 Algorithm analysis .......................................384 10.3 Combinatorial estimates ..................................390 10.3.1 Counting relations, topologies, and partial orders ......394 10.3.2 Generalized Fubini numbers ........................396 10.3.3 The Catalan numbers and binary trees ...............401 10.4 References and comments ................................409 References .....................................................411 List of Symbols ................................................419 Author Index ..................................................423 Subject Index .................................................425 xvi List of Figures 10.3 Monotonie paths of length 2n through an n-by-n grid ........404 10.4 Binary trees .............................................405
adam_txt	Contents Sets and Numbers . 1 1.1 Sets . 1 1.1.1 The concept of a set . 1 1.1.2 Operations on sets . 3 1.1.3 Relations and functions . 6 1.2 Sets of numbers . 11 1.2.1 Two examples . 11 1.2.2 The real number system . 12 1.2.3 Elements of algebra . 20 1.2.4 Elements of topology on Ж . 25 1.2.5 The extended real number system . 30 1.2.6 The complex number system . 31 1.3 Exercises . 32 1.4 References and comments . 40 Vector Spaces and Metric Spaces . 41 2.1 Vector spaces . 41 2.1.1 Finite-dimensional vector spaces . 41 2.1.2 Vector spaces . 44 2.1.3 Normed spaces . 48 2.1.4 Hubert spaces . 49 2.1.5 Inequalities . 51 2.2 Metric spaces . 57 2.3 Compact spaces . 65 2.4 Exercises . 70 2.5 References and comments . 70 Sequences and Series . 73 3.1 Numerical sequences . 73 3.1.1 Convergent sequences . 73 3.1.2 Subsequences . 76 Contents 3.1.3 Cauchy sequences . 77 3.1.4 Monotonie sequences . 80 3.1.5 Upper limits and lower limits . 81 3.1.6 The big Oh and small oh notations . 82 3.1.7 Stolz-Cesaro theorem and some of its consequences . 85 3.1.8 Certain combinatorial numbers . 88 3.1.9 Unimodal, log-convex, and Pólya-frequency sequences . . 94 3.1.10 Some special sequences . 98 3.2 Sequences of functions .108 3.3 Numerical series .110 3.3.1 Series of nonnegative terms .113 3.3.2 The root and the ratio tests .123 3.3.3 Partial summation .125 3.3.4 Absolutely and conditionally convergent series .126 3.3.5 The W-Z method .130 3.4 Series of functions .134 3.4.1 Power series .137 3.4.2 Hyp er geometric series .138 3.5 The Riemann Zeta function ζ(ρ) .139 3.6 Exercises .139 3.7 References and comments .144 Limits and Continuity .147 4.1 Limits .147 4.1.1 The limit of a function .147 4.1.2 Right-hand side and left-hand side limits .152 4.2 Continuity .152 4.2.1 Continuity and compactness .155 4.2.2 Uniform continuous mappings .156 4.2.3 Continuity and connectedness .160 4.2.4 Discontinuities .161 4.2.5 Monotonie functions .161 4.3 Periodic functions .164 4.4 Darboux functions .165 4.5 Lipschitz functions .167 4.6 Convex functions .169 4.6.1 Convex functions .169 4.6.2 Jensen convex functions .173 4.7 Functions of bounded variations .177 4.8 Continuity of sequences of functions .183 4.9 Continuity of series of functions .184 4.10 Exercises .186 4.11 References and comments .189 Contents xi Differential Calculus on К .191 5.1 The derivative of a real function .191 5.2 Mean value theorems .197 5.2.1 Consequences of the mean value theorems .201 5.3 The continuity and the surjectivity of derivatives .206 5.4 L'Hospital theorem .207 5.5 Higher-order derivatives and the Taylor formula .208 5.6 Convex functions and differentiability .214 5.6.1 Inequalities .216 5.7 Differentiability of sequences and series of functions .217 5.8 Power series and Taylor series .219 5.8.1 Operations with power series .222 5.8.2 The Taylor expansion of some elementary functions . 225 5.8.3 Bernoulli numbers and polynomials .228 5.9 Some elementary functions introduced by recurrences .230 5.9.1 The square root function .231 5.9.2 The logarithm function .231 5.9.3 The exponential function .235 5.9.4 The arctangent function .237 5.10 Functions with primitives .239 5.10.1 The concept of a primitive function .239 5.10.2 The existence of primitives for continuous functions . 242 5.10.3 Operations with functions with primitives .244 5.11 Exercises .247 5.12 References and comments .249 Integral Calculus on E .251 6.1 The Darboux-Stieltjes integral .251 6.1.1 The Darboux integral .251 6.1.2 The Darboux-Stieltjes integral .252 6.2 Integrability of sequences and series of functions .262 6.3 Improper integrals .263 6.4 Euler integrals .271 6.4.1 Gamma function .271 6.4.2 Beta function .275 6.5 Polylogarithms .278 6.6 e and π are transcendental .280 6.7 The Grönwall inequality .283 6.8 Exercises .284 6.9 References and comments .287 Contents Differential Calculus on Ж" .289 7.1 Linear and bounded mappings .289 7.1.1 Multilinear mappings .293 7.1.2 Quadratic mappings .294 7.2 Differentiable functions .296 7.2.1 Variations .296 7.2.2 Gâteaux differential .297 7.2.3 Fréchet differential .298 7.2.4 Properties of the Fréchet differentiable functions .300 7.3 Partial derivatives .304 7.3.1 The inverse function theorem and the implicit function theorem .307 7.3.2 Directional derivatives and gradients .312 7.4 Higher-order differentials and partial derivatives .312 7.4.1 The case X = Жп .314 7.5 Taylor formula .316 7.6 Problems of local extremes .317 7.6.1 First-order conditions .317 7.6.2 Second-order conditions .318 7.6.3 Constraint local extremes .319 7.7 Exercises .322 7.8 References and comments .322 Double Integrals, Triple Integrals, and Line Integrals .325 8.1 Double integrals .325 8.1.1 Double integrals on rectangles .325 8.1.2 Double integrals on simple domains .331 8.2 Triple integrals .333 8.2.1 Triple integrals on parallelepipeds .333 8.2.2 Triple integrals on simple domains .340 8.3 и -fold integrals .341 8.3.1 Ti-fold integrals on hyperreet angles .341 8.3.2 n-fold integrals on simple domains .345 8.4 Line integrals .346 8.4.1 Line integrals with respect to arc length .346 8.4.2 Line integrals with respect to axis .347 8.4.3 Green formula .347 8.5 Integrals depending on parameters .349 8.6 Exercises .353 8.7 References and comments .354 Contents xiii 9 Constants .355 9.1 Pythagoras's constant .355 9.1.1 Sequences approaching л/2 . 355 9.2 Archimedes' constant .356 9.2.1 Recurrence relation .356 9.2.2 Buffon needle problem .358 9.3 Arithmetic-geometric mean .358 9.4 BBP formulas .363 9.4.1 Computing the nth binary or hexadecimal digit of π . . 363 9.4.2 BBP formulas by binomial sums .368 9.5 Ramanujan formulas .372 9.6 Several natural ways to introduce number e .374 9.7 Optimal stopping problem .377 9.8 References and comments .378 10 Asymptotic and Combinatorial Estimates .381 10.1 Asymptotic estimates .381 10.2 Algorithm analysis .384 10.3 Combinatorial estimates .390 10.3.1 Counting relations, topologies, and partial orders .394 10.3.2 Generalized Fubini numbers .396 10.3.3 The Catalan numbers and binary trees .401 10.4 References and comments .409 References .411 List of Symbols .419 Author Index .423 Subject Index .425 xvi List of Figures 10.3 Monotonie paths of length 2n through an n-by-n grid .404 10.4 Binary trees .405
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publishDateSort	2009
publisher	Springer
record_format	marc
series2	CMS books in mathematics
spelling	Mureşan, Marian ca. 20./21. Jh. Verfasser (DE-588)1127871331 aut A concrete approach to classical analysis Marian Mureşan New York, NY Springer 2009 XVIII, 433 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier CMS books in mathematics Calculus, Integral Differential calculus Mathematical analysis Differentialrechnung (DE-588)4012252-9 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Integralrechnung (DE-588)4027232-1 gnd rswk-swf Differentialrechnung (DE-588)4012252-9 s Integralrechnung (DE-588)4027232-1 s DE-604 Analysis (DE-588)4001865-9 s Erscheint auch als Online-Ausgabe 978-0-387-78933-0 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754134&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Mureşan, Marian ca. 20./21. Jh A concrete approach to classical analysis Calculus, Integral Differential calculus Mathematical analysis Differentialrechnung (DE-588)4012252-9 gnd Analysis (DE-588)4001865-9 gnd Integralrechnung (DE-588)4027232-1 gnd
subject_GND	(DE-588)4012252-9 (DE-588)4001865-9 (DE-588)4027232-1
title	A concrete approach to classical analysis
title_auth	A concrete approach to classical analysis
title_exact_search	A concrete approach to classical analysis
title_exact_search_txtP	A concrete approach to classical analysis
title_full	A concrete approach to classical analysis Marian Mureşan
title_fullStr	A concrete approach to classical analysis Marian Mureşan
title_full_unstemmed	A concrete approach to classical analysis Marian Mureşan
title_short	A concrete approach to classical analysis
title_sort	a concrete approach to classical analysis
topic	Calculus, Integral Differential calculus Mathematical analysis Differentialrechnung (DE-588)4012252-9 gnd Analysis (DE-588)4001865-9 gnd Integralrechnung (DE-588)4027232-1 gnd
topic_facet	Calculus, Integral Differential calculus Mathematical analysis Differentialrechnung Analysis Integralrechnung
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754134&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT muresanmarian aconcreteapproachtoclassicalanalysis

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